Problem: A circle has a circumference of $10\pi$. It has an arc of length $\dfrac{82}{9}\pi$. What is the central angle of the arc, in radians? ${10\pi}$ ${\dfrac{82}{45}\pi}$ $\color{#DF0030}{\dfrac{82}{9}\pi}$
Solution: The ratio between the arc's central angle $\theta$ and $2 \pi$ radians is equal to the the ratio between the arc length $s$ and the circle's circumference $c$ $\dfrac{\theta}{2 \pi} = \dfrac{s}{c}$ $\dfrac{\theta}{2 \pi} = \dfrac{82}{9}\pi \div 10\pi$ $\dfrac{\theta}{2 \pi} = \dfrac{41}{45}$ $\theta = \dfrac{41}{45} \times 2 \pi$ $\theta = \dfrac{82}{45}\pi$ radians